Download Selection Pressures and Plant Pathogens: Stability of Equilibria

Letter to the Editor
In Response to 'Selection Pressures and Plant
Pathogens: Stability of Equilibria'
K. J. Leonard and R. J. Czochor
Plant Pathologist, Science and Education Administration (SEA), United States Department of Agriculture,
Department of Plant Pathology, North Carolina State University, and Graduate Student, Biomathematics Program,
Department of Statistics, North Carolina State University, Raleigh, NC 27607, respectively.
Cooperative investigations of SEA, USDA and the North Carolina Agricultural Experiment Station.
Journal Series Paper 5621 of the North Carolina Agricultural Experiment Station, Raleigh, NC.
Accepted for publication 3 May 1978.
According to Sedcole (3), Leonard's (1) model of
selection pressures in plant host-pathogen systems does
not fit the observed behavior of these systems in nature.
There might be several reasons for Sedcole's conclusion:
(i) the model contains inaccurate information, (ii) the
model is accurate as far as it goes but it omits some
factor(s) important in nature, (iii) Sedcole misinterpreted
the model, or (iv) Sedcole misinterpreted the behavior of
natural host-pathogen systems.
Sedcole (3) suggested some areas in which he feels that
the model may be inaccurate, but the main basis for his
criticism is his demonstration that the nontrivial
equilibrium point is unstable. This lack of a stable
equilibrium point raises some interesting questions about
the model, but (as shown below) the consequences of
instability of the equilibrium point differ from those he
described.
Before discussing the implications of the unstable
equilibrium point, there are some specific areas of
confusion in Sedcole's (3) interpretation of the model to
be resolved. First, the meaning of the term stabilizing
selection as Sedcole used it is different from its meaning in
Leonard's (1) paper. Leonard used the term stabilizing
selection in the way that Van der Plank (4) described it in
the context of genes for virulence in plant pathogens. In
Van der Plank's use of the term, stabilizing selection is
selection that favors alleles for avirulence, as opposed to
directional selection that favors alleles for virulence,
Sedcole used the term stabilizing selection in its proper
genetical sense. In a single-locus model, stabilizing
selection means the forces that maintain gene frequencies
near their equilibrium values (2). That is, stabilizing
selection would act to reduce frequencies of genes for
virulence when they were above the equilibrium level, but
would act to increase their frequencies when they were
below equilibrium levels. Thus, stabilizing selection
should be represented as the combination of two
opposing forces that lead to a stable equilibrium. Sedcole
is correct in saying that a model without a stable
equilibrium point cannot be an explanation of stabilizing
selection in its proper genetical sense. It should be pointed
out, however, that there is nothing in the model or in
Sedcole's analysis of it that would indicate that there is
not a force of selection that acts against unnecessary genes
for virulence [i.e., stabilizing selection sensu Van der
Plank (4)].
00032-949X/78/000169$03.00/0
Copyright © 1978 The American Phytopathological Society, 3340
Pilot Knob Road, St. Paul, MN 55121. All rights reserved,
To avoid this kind of confusion in the future, plant
pathologists should discontinue current usage of
stabilizing selection in plant pathology as selection
favoring alleles for avirulence on susceptible host plants.
It would be better to refer to this type of selection simply
as selection against unnecessary virulence. As illustrated
in Leonard's (1) model, selection against unnecessary
virulence is essential to the existence of equilibrium points
(whether stable or unstable) at intermediate gene
frequencies.
Sedcole (3) questioned the correctness of Leonard's (1)
assumption that loss of host fitness due to disease is
proportional to the fitness of the pathogen genotype
infecting the host. He suggested that differences in
tolerance among host genotypes would make this
assumption invalid. It seems obvious to us that when
disease damage is proportional to the rate of pathogen
increase, a pathogen genotype of greater fitness should
cause greater disease damage in the host than a less-fit
pathogen genotype would cause. This is all that is implied
in the assumption. A host genotype with more tolerance
may be damaged less than a genotype with less tolerance,
but the assumed relationship between more-fit and less-fit
pathogen genotypes should still hold true. Even on the
tolerant host plant, the pathogen genotype with greater
fitness should cause more damage than a less-fit pathogen
genotype would cause. If significant changes in tolerance
occur as pleiotropic effects of major genes for resistance,
that condition could be added to the model, but there is
no evidence available now to suggest that it should be.
Sedcole (3) suggested that the model assumes that the
entire host population is attacked. This is a
misinterpretation. The model assumes that host plants
are infected at random, so that all plants have an equal
probability of becoming infected, but it is not necessary
that every plant in the host population must be infected.
Of course, an infection will be successful only if the
genotypes of the host and pathogen are compatible. The
probability of a compatible infection depends on the
genetic compositions of the host and pathogen
populations. There is no provision in the model for
protected sites where part of the host population may
grow but the pathogen cannot enter and cause disease.
Such a provision is sometimes included in predator-prey
models.
Sedcole's (3) statements that equations 1 and 3 (in his
letter) make no provisions for changes in host gene
frequencies and pathogen gene frequencies, respectively,
also may be a misinterpretation. In the model the
frequencies of resistant and susceptible plants are
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PHYTOPATHOLOGY
972
assumed to remain constant during the growing season,
but the frequency of genes for virulence in the pathogen
population may change according to forces of selection.
In the next growing season a new host population is
initiated from seeds produced in the preceding season. A
change in frequencies of resistant and susceptible plants
may occur depending on the relative numbers and
viability of seeds produced by each genotype in the
previous season. Seed production by resistant and
susceptible host genotypes depends upon the amount of
disease suffered by each, which in turn is related to the
composition of the pathogen population as well as to the
suitability of the environment for disease increase. The
model assumes that the gene frequencies in the pathogen
population do not change during the interval between
growing seasons, but a factor for differential survival can
easily be added to the model. Thus, under the
assumptions of the model, changes in genetic
composition of host and pathogen populations occur in a
series of alternate steps. During the growing season the
pathogen population adjusts to the host population as it
exists in that season. The new host population in the next
growing season represents an adjustment to the disease
damage caused by the pathogen population of the
previous growing season. Over the course of a year each
population may change, and the change in each
population may cause a complementary change in the
other population. Thus, nj+l = g(pj, nj) and pj+i = f(pj, nj+l).
Sedcole (3) determined mathematically that the
nontrivial equilibrium point in the model at n = (ts c)/ (ts + as) and I - q 2 = k/ (a + t) is not stable. In addition
to the nontrivial equilibrium point, there are also eight
trivial equilibrium points at: n = 0, p = 0; n = 1, p = 0; n=
2
1, p = 1;n=0,p= 1;n=0,1q 2=k/(a+t);n= 1, l-q =
k/(a + t); n = (ts - c)/(ts + as), p = 0; and n = (ts- c)/(ts +
as), p = 1. Using an analysis similar to Sedcole's we have
shown that each of these trivial equilibrium points also is
unstable. This means that if the gene frequencies are
displaced slightly from any of the nine equilibrium points,
they will not return to that equilibrium point. What
exactly they will do is beyond the scope of the analysis
used by Sedcole, because that analysis applies only to the
area in the immediate vicinity of the equilibrium point.
It can be shown mathematically that if host and
pathogen populations are of unlimited size, it is
impossible for n or p to reach 1 or0 within a finite number
of generations unless either n or p is already equal to 1 or
0. In the model,
2
2
2
t) - k]
n + An = n [1 -(1- q )t] + (mnn- n ) [(1 - q ) (a+
1 - (I
q 2 )t + n[(l
-
-
q 2) (a + t)
-
never can be infinitely large it is worth considering this
point, because for reasonably large populations forward
and reverse gene mutation may effectively prevent gene
frequencies from reaching 1 or 0. Thus, the perpetual
cycling of gene frequencies may be a characteristic of
finite as well as infinite populations.
Mathematically defining the behavior of gene
frequencies in this model at values other than the
equilibrium frequencies is an extremely complex
problem. Therefore, Sedcole's (3) use of a numerical
analysis appears to be the best available approach.
However, Sedcole's numerical analysis is incorrect. Using
either of two independently developed computer
programs for the model (one by H. E. Schaffer,
Department of Genetics, North Carolina State University
and one by R. J. Czochor), we find that gene frequencies
do not spiral outward in the way that Sedcole described.
We used the same parameters as Sedcole (a= 0.1, c= 0.01,
k = 0.3, s = 0.2, and t = 1.0) and started at values of no =
0.8626 and p0 = 0.1462 that are within 0.1% of the
equilibrium values. We ran the analysis through 200
iterations and found no detectable outward spiral; the
track of the third cycle around the equilibrium point was
essentially the same as that of the first. Using these same
values except that pa = 0.1329 (at equilibrium P = 0.1472),
we found that n reached a maximum value of 0.907 in the
first cycle and oscillated with a period of about 70
iterations per cycle, but the value of n never exceeded
0.907 at any time even though the analysis was continued
through 3,000 iterations. Again, the last cycle followed
1.0
0.8
0,6-
P
04
02
k]
0.0
=n [-
)t] + n [(I - q ) (a + t)-
(-q
-
2
2
2
k]
2
(1 -q )t + n[(1 - q ) (a + t) - k]
From this it can be seen that if n< 1, n+ An< 1,andifn>
0, n + An > 0. Therefore, if host and pathogen
populations were of infinite size, the frequencies of genes
for virulence and resistance would cycle forever around
the nontrivial equilibrium point unless the frequencies
were originally af equilibrium. Although populations
0.2
0.6
0.4
0.8
1.0
n
Fig. 1. Change in gene frequencies in a theoretical hostpathogen system with n = frequency of the virulence gene, p =
frequency of the resistance gene, no = 0.1 and p0 = 0.6 are initial
parameters exist: cost of
gene frequencies, and theoffollowing
resistance (c) = 0.1; difference in
virulence (k) = 0.3, cost
fitness of the virulent pathogen genotype on the resistant host
compared with the susceptible host (a) = 0.0, effectiveness of
resistance (t) = 1.0, and suitability of the environment for disease
development (s) = 0.2. The nontrivial equilibrium point is
represented by the cross.
July 1978]
LEONARD AND CZOCHOR: LETTER TO THE EDITOR
essentially the same track as the first.
Under some conditions we find an inward spiral from
the starting point toward the equilibrium point. For
example, using the parameters a = 0.0, c = 0.10, k = 0.3, s
= 0.2, and t = 1.0, we started an analysis at n = 0.10 and P
= 0.60. Under these conditions the period of oscillations
varied from more than 150 iterations per oscillation to
less than 90. We found that the gene frequencies spiraled
inward markedly in the first cycle and less in subsequent
cycles(Fig. 1). After more than 900 iterations (nine cycles)
there was still a detectable inward spiral of the gene
frequencies. We did not continue the analysis beyond the
ninth cycle, but obviously the spiral could not continue all
the way to the equilibrium point because the equilibrium
point is unstable. Therefore, there must be a limit cycle
beyond which the inward spiral could not continue. At
this stage in our study of the model we suspect that the
model has many concentric limit cycles so that the
amplitude
ofatgene
frequency
oscillations depends largely
on the point
which
gene frequencies
start in the system.
We do not agree with Sedcole's (3) conclusion that the
model is not consistent with the behavior of gene
frequencies in the 'stable' host-pathogen systems in the
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Middle East. From what we know of these systems we
systems with stable equilibrium points or more typical of
systems that have unstable equilibrium points but stable
limit cycles. Either type of system tends to retain genes for
resistance and virulence in host and pathogen
populations, respectively.
We are exploring many modifications of the model
such as including provisions for gene mutations, variable
weather conditions, etc. and will report the conclusions of
these studies in a later paper.
LITERATURE CITED
1. LEONARD, K. J. 1977. Selection pressures and plant
pathogens. Ann. N. Y. Acad. Sci. 287:207-222.
2. METTLER, L. E., and T. G. GREGG. 1969. Population
genetics and evolution. Prentice Hall, Englewood Cliffs,
New Jersey. 212 p.
3. SEDCOLE,
R. 1978. ofSelection
pressures and plant
pathogens:J. stability
equilibria.
Phytopathology
68:967-970.
4. VAN DER PLANK, J. E. 1968. Disease resistance in plants.
Academic Press, New York. 206 p.